The multidimensional Schr\"odinger operator with a~periodic potential

M. M. Skriganov

Izvestiya. Mathematics , Tome 22 (1984) no. 3, pp. 619-645

Résumé

In this paper the author investigates the zonal structure of the spectrum of the three-dimensional Schrödinger operator with periodic potential. The main result is an estimate of the number $n(\lambda)$ of zones of the spectrum covering the real point $\lambda$. It is shown that, under certain conditions on the period lattice of the potential, $n(\lambda)>\lambda$ when $\lambda\to\infty$. From this estimate it follows that the number of lacunae in the spectrum of the Schrödinger operator is finite. It is also shown that for periodic potentials with small norm there are in general no lacunae in the spectrum. Analogous results are formulated for the Schrödinger operator in higher dimensions. Bibliography: 18 titles.